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<b>let </b><font color="Maroon">F</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<b>assume </b><a NAME="E1:88"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">F</font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Maroon">H</font> &amp; <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Maroon">H</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D41" target="_self">ZF_LANG:def 41</a><br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">L</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:88"/><i><font color="Green">E32</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Maroon">H</font> )
 <b>and </b><br/><a NAME="E4:88"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">H1</font>, <font color="Olive">F1</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <br/>( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">F1</font> &amp; <font color="Olive">H1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Olive">F1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E5:88"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T29" target="_self">Th29</a>, <a class="ref" href="zf_lang.html#T31" target="_self">Th31</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E6:88"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:88"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L1</font> = <font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/>
<b>take </b><font color="Maroon">m</font> = 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D40" target="_self">ZF_LANG:def 40</a><br/>

<b>take </b>
<font color="Maroon">L1</font>
;<br/>

<b>thus </b><a NAME="E11:88"/><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>

<a NAME="E12:88"/>
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E13:88"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T31" target="_self">Th31</a>, <a class="ref" href="zf_lang.html#T33" target="_self">Th33</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>

<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:88_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E2:88_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">j1</font> where <font color="Olive">j1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j1</font> &amp; <font color="Olive">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> ) </span>}</span> 
 ;<br/><a NAME="E3:88_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:88_1"/>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:88"/>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T31" target="_self">Th31</a>, <a class="ref" href="zf_lang.html#T34" target="_self">Th34</a></i>;<br/>

<a NAME="E16:88"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">G1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E18:88"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T32" target="_self">Th32</a>, <a class="ref" href="zf_lang.html#T33" target="_self">Th33</a>, <a class="ref" href="zf_lang.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E19:88"/>
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T31" target="_self">Th31</a>, <a class="ref" href="zf_lang.html#T33" target="_self">Th33</a>, <a class="ref" href="zf_lang.html#T36" target="_self">Th36</a>, </i>;<br/>
<b>hence </b><a NAME="E20:88"/>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T34" target="_self">Th34</a>, <a class="ref" href="zf_lang.html#T35" target="_self">Th35</a>, <a class="ref" href="zf_lang.html#T36" target="_self">Th36</a></i>;<br/>

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E21:88"/><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E22:88"/><b>then </b><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> &amp; 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E23:88"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E24:88"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T33" target="_self">Th33</a>, <a class="ref" href="zf_lang.html#T37" target="_self">Th37</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">F1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a>, <font color="Maroon">G1</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E26:88"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T32" target="_self">Th32</a>, <a class="ref" href="zf_lang.html#T37" target="_self">Th37</a></i>;<br/>
<b>take </b>
<font color="Maroon">F1</font>
;<br/>

<b>take </b>
<font color="Maroon">G1</font>
;<br/>

<b>thus </b><a NAME="E27:88"/>
( <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F1</font> <a href="zf_lang.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T35" target="_self">Th35</a>, <a class="ref" href="zf_lang.html#T37" target="_self">Th37</a>, <a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="zf_lang.html#T39" target="_self">Th39</a></i>;<br/>


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